SlowGuess/NeurIPS_2023_OCR huggingface dataset:
about  $42.28\%$  of the nodes have the degree of 1 in the final network. Those nodes with degree 1 are the key factor that leads to the increase of diameter. Thus, we remove these degree 1 nodes, and calculate the effective diameter of the remaining part. The value is 4.20, which shrinks again. After that, we repeat this process again, and find that it only has  $2.26\%$  nodes with degree 1 in the remaining part. Similarly, we delete those degree 1 nodes, and compute its effective diameter again, which is 4.28 and stays the same as before. This suggests that there is a giant component with extremely high connectivity, and nodes with degree 1 is only a thin layer at the outside of the well-connected giant component. This observation brings new challenges for some downstream tasks.
E More Analyses on Continuous Subgraph Matching
**Frameworks.** We evaluate the performance of six recent CSM frameworks in this section. (1) SJ-Tree [13] proposes a lazy search algorithm, and the search strategy is determined by a vertex-to-vertex basis, which depends on the likelihood of a matching in the vertex neighborhood. (2) IEDyn [30, 31] randomly selects a node from query graph, then it products matching order by conducting DFS on the query graph. (3) SymBi [50] employs a dynamic candidate space as an auxiliary data structure for filtering. (4) Graphflow [33] first generates matching order offline, and then retrieves it in online processing. (5) TurboFlux [35] employs a concise representation of the intermediate results, and a novel edge transition model is proposed to identify the update operations that may affect the current solutions. (6) RapidFlow [68] performs batch subgraph matching via designing a query reduction technique, then dual matching is utilized to leverage the duality of the graph in the matching procedure.
Settings. Similar to the link prediction task, we also remove all the transactions related to the Null address, which gets rid of the impact of the extreme large degree node. In previous studies [69, 35, 13], the initial graph is constructed using the first  $90\%$  of edges, while the rest  $10\%$  of edges are employed as insertion streams. In our scenario, we know the exact time of each transaction, thus we use NFT transactions from year 2017 to the end of 2021 as the initial graph, and then the transactions in the year of 2022 are regarded as the insertion streams. Since the original nodes and edges do not have fine-grained labels, each node is assigned a label randomly selected from a pool of 30 labels. We do not assign labels for edges following the previous works [69, 35]. For evaluation metrics, we report the query time, which denotes the time taken by the online matching procedure to execute. We discard the graph update time, since it's same for all the frameworks. To complete the experiments under an affordable time, a one-hour time limit is imposed for query processing (i.e.,  $3.6 \times 10^{6}$  ms). Additionally, we also calculate the number of matched subgraphs for each query graph.
Results. As we have discussed in Section 4.2, there exist lots of hub nodes in the NFT transaction network. Therefore, we are interested in whether updating the index of hub nodes will have a great impact on the query time. Specifically, except Graphflow, all the remaining frameworks belong to the index-based incremental computation. In this context, the query time consists of the index time, which signifies the time spent on updating the index, and the enumeration time that enumerates the matched results. Therefore, we first sort the node degrees in the descending order, and the results show that the top-50 largest hub nodes have the largest degree with 433,114 and the smallest degree
(a) Query time on pattern 1
(b) Query time on pattern 3
Figure 9: Query time with deleted hub nodes.
Table 8: Temporal link prediction results with live-update settings in sampled subset. We repeat experiments with 3 random seeds to report the mean as well as standard deviation of AUC and MRR. We also present the results under different time snapshot granularities, e.g., days, weeks and months.
Models	Snapshot Days	Snapshot Weeks	Snapshot Months
AUC	MRR	AUC	MRR	AUC	MRR
Dyngraph2vec	77.39±2.04	40.29±5.75	75.33±7.15	36.84±4.97	72.03±2.73	34.83±4.20
TGCN	95.49±0.78	62.40±7.89	91.09±0.46	41.69±7.82	86.13±1.33	43.67±6.23
EvolveGCN	84.34±2.81	40.91±7.96	79.13±0.92	40.72±3.82	78.40±3.33	42.86±3.71
GCRN-GRU	91.41±1.24	33.65±9.48	90.78±2.43	42.88±0.48	82.64±0.22	38.99±0.18
GCRN-LSTM	93.97±1.66	41.63±3.22	90.16±1.83	39.20±7.19	83.38±0.71	37.22±2.63
DynGEM	95.67±0.74	49.25±6.01	93.34±0.98	45.41±5.29	90.75±1.29	41.38±0.61
Roland-MA	97.55±0.59	37.18±6.76	94.59±0.50	44.26±1.26	87.69±2.17	37.98±4.61
Roland-MLP	96.38±1.07	60.44±10.0	89.25±0.89	52.32±3.20	87.89±0.96	42.02±3.37
Roland-GRU	96.52±0.08	69.34±2.68	92.83±0.82	52.74±2.60	91.01±0.77	41.59±2.02
with 7,623. Then, we delete the top-50 hub nodes sequentially and report the query time in Figure 9 for pattern 1 and pattern 3. Note that, we remove the randomly assigned node labels in this situation, which counts the query time associated with the hub nodes. As can be seen, the query time almost remains the same with acceptable fluctuations across all the frameworks except for RapidFlow. This is because RapidFlow is extremely efficient (i.e., only taking several hundred milliseconds), thus a slight fluctuation will be very obvious in the figure. We can conclude that these four frameworks are applicable to graphs with dense structures. They can effectively serve as a bridge to generate ground truth for continuous subgraph matching, providing valuable support for the training of deep graph learning models.
F Temporal Link Prediction Results on Sampled Subset
To fully evaluate the models' performance, we also sample a subset of data spanning from January 1st 2020 to August 31st 2020. This subset contains 88,112 nodes and 203,221 edges. We provide the temporal link prediction results under live update setting in Table 8. As can be seen from the results, we can draw similar conclusions. Notably, we also observe that with the increase in time granularity, the performance of the models exhibits a corresponding decline within this sub-sampled dataset.
G TEA and TET Plots
Following the definitions in [57], we present Temporal Edge Appearance (TEA) and Temporal Edge Traffic (TET) plots in Figure 10. Specifically, a TEA plot visualizes the proportion of recurring edges compared to newly joined edges at each timestamp within a temporal graph. In Figure 10a, the gray bar represents the count of edges that are previously observed, while the red bar signifies the quantity of new edges generated at each subsequent time step. The TEA plot illustrates that our
(a) TEA plot with day intervals
Figure 10: TEA and TET plots.
(b) TET plot with sampled 2 million edges
dataset exhibits a substantial proportion of newly formed edges at each time step, which means the transaction graph is highly active. In contrast, a TET plot represents the recurrent pattern of edges across different time intervals. Edges are colored to indicate whether they appear solely in the training set (green), solely in the test set (red), or in both sets (orange). As it is time consuming to plot all the 124 million edges in the figure, we randomly sample 2 million edges and show the results in Figure 10b. We have tried to use all the data, but the plotting process was unable to complete within 12 hours. The sampled 2 million edges subset showcases a subtle recurrence pattern, which is consistent with our analyses in previous sections.